Maximizing Total Wealth Implies that the return of the Marginal Investment is Negative

Suppose you have k possible assets to invest in. For the sake of this argument, we will make two assumptions. First that each investment has a different default rate r of appreciation. Secondly we will assume that each investment has a limited capacity. Once the limit is reached, it is not possible to invest more.

For this specific example consider 10 investment assets, that range in returns from -3% to +7%. We will assume that each has a capacity of $200 million, in today's dollars. (as the investments appreciate, the capacity increases at the same rate).

How will these investments be chosen?

Well, it is really quite simple. The highest yield investment will be preferred until it is saturated at capacity. Then the next highest yield, etc.

So if the total wealth of all investment assets is $1 billion, or 5x $200 million, then the last dollar invested will have a return of 3%, and the next dollar will have a return of just 2%. The average return will thus be 5%.

If the total wealth of all investments is $2 billion, or 10x $200 million, then even our negative yielding assets have been developed to capacity. So the rate on the last dollar invested will be -3%, the average rate will be 2%, and there will be no investment opportunity for the marginal dollar or unit of labor.

If we want to actually figure out what the discount rate will be, that will depend on the time horizon. The easiest is just a time horizon of 1 year. Then we just do a weighted average with 0.97 at -3%, 0.98 at -2%, 1.00 at 0%, 1.01 at 1%, etc.